If a card deals X damage to a card at the battlefield, but I chain a card that moves it from the battlefield, would that damage still apply?

Ruling: No, the damage does not apply if the unit is moved from the battlefield before the damage resolves. When a spell targets a unit at a specific location, that unit must be at that location both when targeted and when the effect resolves.

Sequence:

• The spell targets a unit at a specific location (the battlefield)
• If the unit is not at a valid location when targeting, the spell does not enter the chain
• On resolution, if the target is no longer valid (moved from the battlefield), the "do as much as you can" rule applies
• The damage does not occur because the unit is no longer at the required location

Nuances:

• If the effect depends on the location of another unit (like "here"), moving either unit involved would prevent the effect from happening
• Some spells like Hextech Ray target the unit itself rather than the battlefield location, and will deal damage as long as the unit is at any battlefield when it resolves